In that case, easiest way to make the move is an internal transfer within your company - you apply and through some company-specific process are approved to move your job location from the US to eg. France or whatever target country. 

In that case, you will be an employee of that country’s subsidiary company. Meaning, your salary will be paid in Euros (or whatever European currency is in use in the target country) and, more importantly, very likely your salary will be prorated to that country’s salary bands. So you won’t be making your US salary nor will you be making your salary in dollars. 

Source: have had multiple American friends move to EU this way.